奇诱导子图与奇偶诱导子图的界
Bounds on Odd and Odd-Even Induced Subgraphs
AI总结：
该研究针对无孤立顶点的图，推广奇割方法改进奇偶容许集下界，推导二分图奇容许集的更优下界，构造二分图确定对数加性改进的最优量级。
AI中文摘要：
设G为n顶点图，ℓ:V(G)→F₂指定顶点的度数奇偶性。集合S⊆V(G)是ℓ-容许的，当且仅当对每个v∈S，v在诱导子图G[S]中的度数与ℓ(v)模2同余。记h_ℓ(G)为ℓ-容许集的最大阶，定义f_oe(G):=min_ℓ h_ℓ(G)，f_o(G):=h_1(G)，其中1(v)=1对所有v∈V(G)成立。我们针对无孤立顶点的图证明三个主要结果：第一，通过将Zeng的奇割方法推广到任意奇偶指定并引入单侧引理，证明对任意ℓ有h_ℓ(G)≥n/6，因此f_oe(G)≥n/6，改进了先前的界2n/21；第二，对二分图，我们基于二分邻接矩阵的F₂秩推导f_o(G)的下界，合并后得到f_o(G)≥(1/4+1/256)n=65n/256，因此二分图情况下，Scott界f_o(G)≥n/(2χ(G))中的因子2可替换为128/65<2；最后，记α=α(G)，四阶矩论证给出当α≥2时，f_o(G)≥α/2 + (log₃α)/8 - (1/4)log₃log₃√α。我们还构造了满足f_o(G)≤α(G)/2 + log₂(α(G)+1)+1/2的二分图，表明对Scott界f_o(G)≥α(G)/2的对数加性改进具有最优量级。
英文摘要：
Let $G$ be an $n$-vertex graph and let $\ell:V(G)\to\mathbb{F}_2$ prescribe degree parities. A set $S\subseteq V(G)$ is $\ell$-admissible if every $v\in S$ has degree congruent to $\ell(v)$ modulo $2$ in $G[S]$. Let $h_\ell(G)$ be the maximum order of an $\ell$-admissible set, set $f_{\mathrm{oe}}(G):=\min_\ell h_\ell(G)$, and write $f_o(G):=h_{\mathbf{1}}(G)$, where $\mathbf{1}(v)=1$ for every $v\in V(G).$ We prove three main results for graphs without isolated vertices. First, by extending Zeng's odd-cut method to arbitrary parity prescriptions an introducing a one-sided completion lemma, we show that $h_\ell(G)\ge n/6$ for every $\ell$. Consequently, $f_{\mathrm{oe}}(G)\ge n/6$, improving the previous bound $2n/21$. Second, for bipartite graphs we derive lower bounds on $f_o(G)$ in terms of the $\mathbb{F}_2$-rank of the bipartite adjacency matrix and combine them to obtain \[ f_o(G)\ge \left(\frac14+\frac1{256}\right)n=\frac{65}{256}n. \] Thus, in the bipartite case, the factor $2$ in Scott's bound $f_o(G)\ge n/(2χ(G))$ can be replaced by $128/65<2$. Finally, writing $α=α(G)$, a fourth-moment argument gives, for $α\ge2$, \[ f_o(G)\ge \fracα{2}+\frac{\log_3α}{8} -\frac14\log_3\log_3\sqrtα. \] We also construct bipartite graphs satisfying \[ f_o(G)\le \frac{α(G)}2+\log_2\!\bigl(α(G)+1\bigr)+\frac12, \] showing that the logarithmic additive improvement over Scott's bound $f_o(G)\geα(G)/2$ has the optimal order of magnitude.
